Equilibrium Impact of Value-At-Risk
∗
Markus Leippold†
University of Zürich
Fabio Trojani
‡
University of Southern Switzerland
Paolo Vanini
§
University of Southern Switzerland
This Version: January 31, 2003
First Version: December 14, 2001
∗
The authors thank seminar participants at the University of Zürich, the Swiss Federal Institute
of Technology Zürich, the University of Southern Switzerland, the Bendheim Center of Finance at
Princeton University, the Quantitative Methods in Finance 2002 Conference, and the 9th Symposium
on Finance, Banking, and Insurance in Karlsruhe. In particular, we thank Giovanni Barone-Adesi,
Damir Filipovic, Rajna Gibson, Michel Habib, Ronnie Sircar, and Liuren Wu for their valuable comments. Fabio Trojani and Paolo Vanini acknowledge the financial support of the Swiss National Science
Foundation (NCCR FINRISK). We welcome comments, including references to related papers we inadvertently overlooked.
†
Correspondence Information: Markus Leippold, Swiss Banking Institute, University of Zürich,
Plattenstrasse 14, 8032 Zürich, Switzerland, tel: (01) 634 39 62, mailto:[email protected]
‡
Correspondence Information: Fabio Trojani, Institute of Finance, University of Southern
Switzerland, Via Buffi 13, 6900 Lugano, Switzerland, mailto: [email protected].
§
Correspondence Information: Paolo Vanini, Institute of Finance, University of Southern
Switzerland, Via Buffi 13, 6900 Lugano, Switzerland, mailto: [email protected].
Abstract
We analyze Value-at-Risk based regulation rules and their effects on financial markets. Our
model is formulated in a continuous-time economy where investors maximize expected utility
subject to some regulatory Value-at-Risk constraint when asset price dynamics exhibit stochastic volatility. We show that in partial equilibrium the effectiveness of VaR regulation is closely
linked to the “leverage effect”, the tendency of volatility to increase when prices decline. We
then extend our analysis to a pure exchange economy and explore the implications of VaR
regulation on equilibrium quantities such as interest rates and volatilities. Analysis of the
general equilibrium model with heterogenous investors indicates ambiguous economic effects.
Nevertheless, we identify a tendency for lower interest rates and higher risk premia. Moreover,
regulation leads to changes in the cross-sectional wealth distributions, which are independent
of the banks’ risk aversion. Summarizing, it is not possible to make a clear statement whether
Value-at-Risk as an endogenous risk measure supports the aim of regulation. In order to maintain the power of analytical expressions, different approximations are necessary in the model.
Particular care is given to control the approximation errors either analytically or numerically.
JEL Classification Codes: G11, G12, G28, D92, C60, C61.
Key Words: Value-at-Risk, Regulatory Policy, Dynamic Financial Equilibria, Perturbation
Theory, Stochastic Volatility, Monte Carlo.
Regulation aims at maintaining and improving the safety of the financial industry. At
the root of every regulation framework lies the basic idea of calculating capital reserves as a
function of the firm’s total capital and its risk. It might be tedious, but straightforward to
calculate the firm’s capital. However, a less obvious task is the measurement of risk. In the 1996
Amendment of the Basle 1988 Accord, the Bank for International Settlement (BIS) extended
their regulatory framework from credit risk to market risk. Regulators advocate normative and
simplified rules applicable to all financial institutions. However, the Amendment recognizes
the complexity of correctly assessing a risk figure to market risk and allows banks, satisfying
some predefined qualitative standards, to use their internal models for regulatory reporting.
In 1996, Value-at-Risk (VaR) based risk management had already emerged as common market
practice. Thus, the BIS chose VaR as standard tool for regulatory reporting. In this paper,
we explore the impacts of VaR based regulation, both in partial and general equilibrium, in
economies with a stochastic opportunity set where stock returns exhibit stochastic volatility.
To our knowledge, Basak and Shapiro (2001) were the first to analyze the equilibrium
impact of VaR constraints on asset prices in a model with continuous trading. In their model,
VaR constrained investors take on a larger risk exposure than unconstrained investors, implying
a deepening and prolongation of market downturns. This unappealing effect can be attributed
to the the lack of subadditivity of VaR. Indeed, adopting a coherent risk measure, Basak and
Shapiro (2001) show that the above deficiency can be removed. Finally, they extend their
model to a general equilibrium populated by homogenous log-investors. Closely related to
Basak and Shapiro (2001) is the work by Berkelaar, Cumperayot, and Kouwenberg (2002). In
a similar setup, they find that the presence of risk managers tends to reduce market volatility,
as intended by regulators. Only in very bad states of the economy, risk managers would adopt
a gambling strategy such that volatility is increased. They further explore the effects of VaR
regulation on volatility smiles.
Cuoco, He, and Issaenko (2001) attribute the results of Basak and Shapiro (2001) to their
static definition of the VaR constraint. In Basak and Shapiro (2001), banks have to report
their VaR to the regulatory authority only at some specific date, while they are allowed to
1
continuously trade before and after that reporting date. This is clearly not consistent with
the current regulatory framework, where banks need to calculate and adjust their VaR at least
daily. Cuoco, He, and Issaenko (2001) extend the Basak and Shapiro (2001) model by letting
the VaR limit be dynamically updated. Assuming geometric Brownian motion for asset price
dynamics Cuoco, He, and Issaenko (2001) show that neither VaR nor Expected Shortfall based
regulation induce riskier portfolio policies in partial equilibrium. However, they do not provide
a general equilibrium analysis, nor do they extend their model to non-normal asset returns.
Danielsson and Zigrand (2001) analyze the general equilibrium effects of VaR constraints in
a static economy and find that VaR regulation adversely affects prices, liquidity and volatility
relative to a benchmark unregulated economy. In Danielsson, Shin, and Zigrand (2001) this
analysis is extended to a multiperiod but still myopic setting, where it is shown by simulation
that VaR exacerbates market shocks1 .
Our model is formulated in continuous time: Investors derive utility both from intermediate
consumption and terminal wealth. In partial equilibrium, we compute optimal policies under
VaR constraints, when stock price drift and volatility depend on a stochastic state variable.
Therefore, our model is a fully dynamic intertemporal choice model which exhibits stochastic
volatility. The non-constancy of volatility is well known in practice especially when markets
are under pressure. Since volatility essentially drives VaR, it is in such stress situations where
risk regulation is likely to become effective by risk figures binding at the risk limits. However,
the price to pay when analyzing a model that comes closer to reality regarding its underlying
assumptions about price dynamics, is the impossibility of obtaining fully analytical solutions.
Since we nevertheless insist on analyticity we rely on approximation procedures. Besides
the approximations, which gives us exact comparative static results, we carefully analyze the
errors and higher order corrections: In particular, we use an Itô-Taylor approximation of
VaR and consider the error bounds analytically and for different input parameters. For the
optimal policy we rely on perturbation theory (see Kogan and Uppal (2002), Trojani and Vanini
(2001)). In this approach, solutions can be constructed to arbitrary precision around the closedform optimal policies of the log-utility investor. In partial equilibrium higher order terms are
2
derived and convergence under a-priori bounds is proven. The accuracy of the policies up to
second order are compared to numerical solutions obtained by Monte Carlo simulations (see
Detemple, Garcia, and Rindisbacher (2003) and Cvitanic, Goukasian, and Zapatero (2003)). A
basic finding is the link between the effectiveness of VaR regulation to the ”leverage effect”, the
tendency of volatility to increase when prices decline. According to the sign of the correlation
between the risky asset and its volatility, the VaR constraint reduces or raises the risk exposure.
If the correlation is negative, investment is reduced before the risk limits are reached due to
the VaR constraint. Therefore, in this case incentives of the bank are not distorted. Such
a negative correlation was found for example by Anderson, Benzoni, and Lund (2002) for
S&P daily returns. Under the normality assumptions of Cuoco, He, and Issaenko (2001)
the interrelationship between asset prices and volatility remains hidden. By “distortion of
incentives”, we mean that, taking the regulator’s perspective, the presence of regulation has a
destabilizing effect on the financial system by increasing the default probability of the financial
institutions. In our paper, we directly connect such an increase in default probability to the
bank’s exposure in the risky asset. If, compared to the unregulated economy, regulation leads
to an increase in the bank’s exposure and to a suboptimal policy for the bank, we say that
regulation distorts incentives.
After having analyzed our model in partial equilibrium, we embed the model in a general equilibrium with heterogenous agents and explore the effects of VaR based regulation on
equilibrium interest, asset price dynamics, and portfolio policies. Hence, VaR becomes an endogenous measure of risk. From this perspective, we follow the general equilibrium analysis in
Basak and Shapiro (2001), imposing more realistic assumptions on a) the VaR measure used,
b) the price dynamics, and c) allowing the economy to be populated by heterogeneous agents
with risk-aversion different from log-utility. A main conclusion is the ambiguity of the results
with respect to any incentive distortions. In other words, no decisive answer can be given in
the models under consideration whether incentives of the financial institution are distorted or
not when they are exposed to VaR constraints. In general, only tendencies can be deduced
from our analysis. This is certainly not a result the financial industry and its regulators await,
3
as the implementation of industry-wide regulation standards is very costly. One would rather
expect to have a clear-cut knowledge about the outcome of applying such standards to the
financial sector.
There is more than one reason for the ambiguous results. First, rational general equilibrium
problems with stochastic volatility are hard to solve per se. To this complexity VaR adds even
more. The simplicity of VaR when used with exogenous investment strategy turns into a
difficult to a hardly-tractable problem if the strategies are the outcome in equilibrium. The
implicit definition of VaR and its non-linearity under stochastic volatility is not well-suited to
test for the sensitivity of regulation on economic variables. More specifically, for different model
specifications under different parameter constellations we find that for a low rate of economic
growth, there is a tendency for VaR regulation to decrease interest rates and to increase
volatility as well as the banks’ exposure to the risky asset. Finally, the heterogenous results
follow with minimal heterogeneity assumptions on the banks. More precisely, the institutions
differ only with respect to their initial wealth and the risk aversion. This contrasts the often
heard, but never proved, argument that uniformity of the regulation can pronounce systemic
crisis. Furthermore, if the risk heterogeneity is removed and the population of institutions is
split into a regulated and not regulated one, the VaR regulation still has an impact on the
homogeneous economy: Wealth is shifted from the regulated to the unregulated institutions.
This is surprising, since the VaR limits are chosen such that the explicit risk constraints are
wealth level independent. This shows that VaR as an endogenous risk measure has an effect on
economic variables in equilibrium which is not apparent in the formulation of the risk measure.
We remark that the framework offered in this paper is not only suitable to analyze the
effect of VaR regulation, but also would allow to analyze the consequences of VaR limits for
traders within a large bank. If traders can move the markets, VaR becomes an endogenous
risk measure. VaR limited traders would try to move the markets in a way that might not
yield the most favorable outcome for the bank.
The paper is structured as follows. The next section introduces the basic model setup.
Section ?? derives the VaR constraint and its approximation. Section 2 elaborates on the
4
bank’s optimization problem and presents its solution in the presence of VaR constraints.
Section 3 investigates the extension to general equilibrium with VaR constrained investors
and derives the implications on interest rates, volatilities, and portfolio policies. Section 4
concludes.
1
The Model
The financial market consists of a risky asset with price Pt and an instantaneous risk-free
money-market account with value Bt . The dynamics of Bt and Pt are
dBt = r(Xt )Bt dt, B0 = 1,
(1)
with r(Xt ) the risk-free rate process, and
dPt = α(Xt )Pt dt + σ(Xt )Pt dZt , P0 = p.
(2)
Uncertainty is modelled by a two dimensional Brownian motion (Zt , ZtX ) having correlation
E[dZt dZtX ] = ρdt. Drift and diffusion parameters in the asset’s dynamics depend on a onedimensional state variable Xt . The process Xt also follows an Itô diffusion
dXt = µX (Xt )dt + σX (Xt )dZtX , X0 = x.
(3)
In the sequel, we split the drift of the asset price process into the short rate component r(Xt )
and a risk premium component λ(Xt ), i.e. α(Xt ) = r(Xt ) + λ(Xt ).
We consider a bank selecting a portfolio fraction wt (1 − wt ) of current wealth Wt invested
in the risky asset (riskless asset). Thus the bank’s wealth dynamics are
dWt
= (wt λ(Xt ) + r(Xt ))dt + wt σ(Xt )dZt .
Wt
(4)
The regulator is supposed to define the bank’s constraint on market risk by means of a
5
VaR risk measure2 .
Definition 1. The time-t VaR of a portfolio wt given P-probability level ν ∈ (0, 1) and for a
fixed time-horizon τ > 0 is defined by
VaRν,w
= inf{L ≥ 0|P (Wt − Wt+τ ≥ L| Ft ) < ν},
t
(5)
where Wt+τ is the portfolio value at time t + τ of a fixed-weight strategy with initial weight wt
at time t.
Definition 1 is consistent with the VaR concept adopted by regulators to measure market
risk. For reporting purposes the time-horizon τ is chosen to be 1 day or 10 days.
In our model, the bank’s VaR is bounded at time t by an exogenous limit VaRt for the
given time-horizon τ . In the sequel we work with a VaR limit VaRt proportional to current
wealth3 , i.e.,
VaRt (W, t) = βWt , β ∈ [0, 1] .
(6)
The wealth dynamics (4) depend on the stochastic opportunity set Xt . Thus, we cannot
expect to obtain generally closed-form solution for the bank’s intertemporal decision problem
in the presence of VaR constraints. To retain analytical tractability, we therefore carefully
approximate the VaR constraint implied by (5), (6) and we apply the Itô Taylor formula to
define the first order approximation,
log Wt+τ
µ
¶
1
(1)
≈ log Wt+τ = log Wt + r(Xt ) + wt λ(Xt ) − wt2 σ(Xt )2 τ.
2
(7)
The accuracy of the approximation (7) is quantified by the next proposition.
(1)
Proposition 1. The approximation error of the first-order approximation Wt+τ for the value
6
Wt+τ of a fixed-weight portfolio with initial weight wt is bounded by
¯
¯ ´
³¯
1
¯
¯
¯
(1)
P ¯log Wt+τ − log Wt+τ ¯ ≥ M ¯ Ft ≤
E [|R|] ,
M
where
¯Z
¯
E [|R|] = ¯¯
t+τ
t
Z
t
s
(8)
¯
¸
¯
1 2
2
E Lr(Xu ) + wt Lλ(Xu ) + wt Lσ(Xu ) |Ft duds¯¯ ,
2
·
2
∂
2 ∂
and L = µX ∂X
is the infinitesimal generator of X.
+ 12 σX
∂X 2
Proposition 1 can be interpreted as follows: P(·|Ft ) is the conditional probability that
the logarithmic difference between the approximated wealth and the true wealth exceeds a
prespecified value M at time t + τ . If we choose, say, M = 0.10, Proposition 1 offers a bound
(1)
on the probability that log Wt+τ and log Wt+τ differ by more than 10%. The accuracy of the
VaR approximation (8) is illustrated in Table 1, where we assumed a mean-reverting geometric
Brownian motion for the volatility process. In this case, E [|R|] can be computed in closed
form. The results are presented for two different conditioning values of the state variable,
Xt = 1 and Xt = 3. For lower values of Xt the approximation is even more accurate. Table
1 shows that the approximation bounds are generally very tight. For instance, the bound on
the error probability for M = 1, X = 1 and an horizon τ = 10 is always below 0.0001. The
bounds for X = 3 are always below 0.01 and in many cases below 0.0005.
The quality of the above approximation results suggests that an approximate VaR calculation based on (7) could be reasonably used to investigate the theoretical properties of portfolio
selection under VaR constraints. In excess of this, the common market practice, where VaR
figures are reported based on a conditional normal distribution assumption, further motivates
our approach.
The advantage of using (7) for computing an approximate VaR constraint is that it implies
some direct portfolio bounds on the optimal policy of the VaR constrained bank. These are
given in the next proposition.
Proposition 2. To first order, the constraint VaR ≤ VaR is equivalent to the following upper
7
and lower bound on the fraction wt of wealth invested in the risky asset,
wb− (Xt ) ≤ wt ≤ wb+ (Xt ),
(9)
where
wb± (X) =
±
v
λ(X)
√
+
2
σ(X)
σ(X) τ
q
√ 2
(λ(X)τ + σ(X) τ v) + 2σ(X)2 τ (r(X)τ − log(1 − β))
σ(X)2 τ
(10)
,
with v = N −1 (ν) the ν-quantile of the standard Normal distribution.
The functional form (6) for the VaR limit implies a bound on the optimal portfolio fraction
that is wealth independent4 .
The basic structure of the bounds in (10) corresponds to a discrete time confidence interval
for wt . The interval is centered at the point
√
λ(X)τ + σ(X) τ ν
.
σ(X)2 τ
This entity can be interpreted as the optimal policy of a log-investor that computes expected
excess returns based on a Gaussian VaR threshold return for the time horizon τ . By inspection
of equation (10), we have
wb+ (X) ≥ 0,
wb− (X) ≤ 0,
for any X. This holds for all functional forms λ(X) and σ(X). Hence, when volatility is an
increasing function of X we can expect the portfolio bounds wb± to tend to ±∞ as X → ∞.
In the next sections we will make use of some polynomial functional forms
λ(X) = λXtn1
,
σ(X) = σXtn2
,
r(X) = rXtn0
,
where λ, σ, r > 0 and n0 , n1 , n2 ∈ N in order to obtain some more explicit characterizations of
8
the impact of VaR constraints on the optimal policies of a VaR constrained investor; cf. also
Assumption 2 below. For these model settings, the parameters n0 , n1 , and n2 will have to
be selected with some care in order to avoid trivial settings where VaR constraints are either
vacuous or binding a priori.
2
Partial Equilibrium
This section is devoted to the analysis of VaR constrained optimal policies in a partial equilibrium framework.
2.1
Optimal Policies
To define the bank’s partial equilibrium optimization problem under VaR constraints we start
from the following standard assumption on preferences.
Assumption 1. The bank derives utility from final wealth WT according to a CRRA-utility
function
u(W, T ) =
Wγ − 1
, γ < 1,
γ
where δ > 0 is a time preference rate.
According to Assumption 1, the bank derives utility from terminal wealth WT only. The
model will be extended in Section 2.4 to allow also for the presence of intermediate consumption
streams.
The major goal of this section is to investigate the partial equilibrium impact of VaR
constraints in the presence of a stochastic opportunity set. Detailed characterizations of these
issues are derived under the following parametric assumptions on X 0 s dynamics, on λt , σt and
rt .
9
Assumption 2. Xt follows a mean reverting process given by
dXt = (θ − κXt )dt + σX Xtm dZtX ,
where κ, θ, σX are positive constants. The risk premium, thw risky asset volatility and the
interest rate processes are of the form
λ(Xt ) = λXtn1
,
σ(Xt ) = σXtn2
,
r(Xt ) = r
,
where λ, σ, r, m, n1 , n2 are positive constants.
Notice that Assumption 2 comprises model settings where volatility is stochastic or where
variance-in-mean effects are present, for instance when n1 6= 0, n2 6= 0 and m 6= 0. On the
other hand, when n1 = 0 we can investigate the impact of VaR constraints under a pure
stochastic volatility setting. In the case n1 = 2n2 , the unconstrained Merton policy of a
log-utility investor is a constant
λ(Xt )/σ(Xt )2 = λ/σ 2
.
Therefore, in this case VaR constraints are either vacuous or binding a priori. Further, when
restricting the parameter m to m = 0, m =
1
2
or m = 1 we obtain a model class that
encompasses the most popular stochastic volatility models, like the lognormal, the squaregaussian, or the square-root model of Heston (1993), and at the same time allows for closed
forms of all moments and cross-moments of Xt . Existence of such closed form expressions
are necessary to obtain analytical asymptotic characterizations of the relevant optimal policies
and equilibrium quantities in the presence of VaR constraints. Generally, computation of the
moments of Xt can be achieved by means of the next technical result.
¤
£ k
|Ft , k ≥ 1, solves the differential
Lemma 1. Under Assumption 2 the k-th moment E Xt+τ
10
equation
£ k
¤
µ h
i
h
i k−1
h
i¶
dE Xt+τ
|Ft
k−1
k−2+2m
k
2
= k θE Xt+τ
|Ft − κE Xt+τ
|Ft +
σX
θE Xt+τ
|Ft
,
dt
2
£
¤
subject to E Xtk |Ft = Xtk .
We now define the VaR-constrained portfolio optimization problem of our bank, based
on the approximate VaR constraint implied by Proposition 15 . Denoting by BR(W, X) the
budget constraints (3), (4) and by C(X) the approximate VaR constraint (10), the optimization
problem of a VaR constrained investor can be written as
(P1) :
J(W, X, t) =
max
w∈R(W,X)
E [u(WT , T ) |Ft ] ,
where R(W, X) = {w ∈ BR(W, X)} ∩ C(X). Intuitively, the solution of problem (P1) has
to provide optimal investment strategies characterized by a region where the VaR constraint
binds and a region where it does not bind. This is the content of the next proposition.
Proposition 3. Consider the control problem (P1) under the Assumptions 1 and 2. In a
region where the J-function is increasing and jointly strictly concave in W and X, the policy
solving (P1) is



w+ (X, t), if wf (X, t) ≥ wb+ (X, t),


 b
w∗ (X, t) =
wb− (X, t), if wf (X, t) ≤ wb− (X, t),




 wf (X, t), else,
where
wf (X, t) = −X n1 −2n2
λJW
2
σ W JW W
− ρX m−n2
σX JW X
,
σW JW W
(11)
and wb± is given in (10).
Apparently, inserting the optimal policies back into the HJB equation for problem (P1)
leads to a non-explicitly solvable PDE. We therefore apply an asymptotic analysis that computes the value function and the implied optimal policies as a power series in γ.
11
2.2
Perturbation with respect to Risk Aversion
From the homogeneity properties of problem (P1), the value function J has to be of the form
J(W, X, t) =
eγg(X,t) W γ − 1
,
γ
(12)
with g(X, t) an unknown function. Expanding the function g(X, t) in the parameter γ, i.e.
1
g(X, t) = g0 (X, t) + γg1 (X, t) + γ 2 g2 (X, t) + O(γ 3 ),
2
(13)
the optimal policy up to first order in γ is easily obtained from (11), (12), and (13) as
(1)
wf (X, t) = (1 + γ)Xtn1 −2n2
ρσX ∂g0 (Xt , t)
λ
+ γXtm−n2
.
2
σ
σ
∂X
(14)
Therefore, if g0 (X, t) can be determined, the first-order policy of an investor with risk aversion
1 − γ is also determined. Equation (14) gives the approximate optimal policy as a sum of two
approximate subpolicies. A myopic portfolio
(1 + γ)Xtn1 −2n2
λ
σ2
and an intertemporal hedging position
γXtm−n2
ρσX ∂g0 (Xt , t)
σ
∂X
.
As usual, the difficulty in characterizing the optimal portfolio solution under a stochastic
opportunity sets consists precisely in determining the optimal intertemporal hedging policy.
Moreover, characterizing the hedging policy of a VaR constrained investor is a crucial issue
in order to understand the equilibrium impact of VaR constraints, as we show below, . Since
we want to analyze the impact of VaR constraints (i) under quite general model settings and
(ii) for several dynamic specifications of the economy, we make use of a perturbation approach
that allows us to analyze the qualitative impact of VaR constraints for a quite broad range of
12
models.
In the perturbation approach, the problem of characterizing the intertemporal hedging
policy is easily circumvented as soon as the function g0 can be computed. By construction,
the unknown function g0 is fully determined as soon as we can compute the value function of
a log-utility investor. This is the basic idea behind the next proposition, which gives a way
to compute g0 as a conditional expectation of some random variables that involve only some
known functions of the future states of the process (Xt ).
Proposition 4. Consider the optimization problem (P1). The function g0 in (14) reads6
g0 (X, t) =
g0f (X, t)
σ2
−
2
Z
1 λ2
= r(T − t) +
2 σ2
λ
f
wlog
(X, t) = Xtn1 −2n2 2 ,
σ
g0f (X, t)
·
T
E
t
Z
T
t
ŵb± (X, t) = wb± (X, t) − Xtn1 −2n2
¯
ll{|wf |>|w± |} (ŵb± (X, s)Xsn2 )2 ¯Ft
log
b
¸
ds,
(15)
h
¯ i
E Xs2(n1 −n2 ) ¯Ft ds,
λ
,
σ2
f
where g0f is the function g0 prevailing in the absence of VaR constraint, and wlog
(X, t) is the
uncontrained optimal portfolio policy of a log utility investor.
With Proposition 15 we are now in a position where a computation of the expectations
in g0 and g0f immediately provides a first order approximation to the optimal portfolio of a
VaR constrained investor for the class of models defined in Assumption 2. The quality of these
approximations can be improved by considering higher order terms in the perturbation series.
Before studying in detail the partial equilibrium effects of VaR constrains, we show in the next
section how such higher order approximations can be obtained and how convergence to the
underlying solution can be achieved under appropriate assumptions.
2.3
Accuracy and Convergence of the Perturbation Approach
Before analyzing the impact of VaR constrains in more detail, we illustrate for an explicit
model setting the accuracy of the above perturbative approach. We proceed in three steps.
13
First, we show how higher order corrections for the optimal policies of problem P1 can be
obtained. Second, we demonstrate that for some given ”reasonable” a priori bounds on the
terms in the perturbation series the whole series converges. Third, we compute different finite
order approximations and investigate in a numerical example the speed of convergence of the
perturbation series. Specifically, we apply Lemma 1 to calculate the function g0f (X, t). In order
to calculate g0 (X, t), we have generally to resort to numerical methods.
To analyze the accuracy of the first order approximations in the last section we first consider
some higher order policy approximations. From (11), (12), and (13), the n-th order policy
approximation is easily obtained as
(n)
wf (X, t)
=
Xtn1 −2n2
n−1
X γ j+1 ∂gj (X, t) 1 − γ n−j
λ 1 − γ n+1
m−n2 ρσX
+
X
.
t
σ2 1 − γ
σ
j!
∂X
1−γ
(16)
j=0
Some natural a-priori bound of on the unknown functions gj (X, t), j ∈ N, which ensure convergence of the perturbation theory to the correct limit policy are given in the next result.
(n)
Corollary 1. Consider the control problem (P1). The optimal policy wf
VaR constraint up to order n converges toward wf if
∂gj (X,t)
∂X
for a non-binding
2j(n1 −n2 )
≤ KXt
, K > 0, j =
0, . . . , n. The limit policy is bounded by
2(n1 −n2 )
wf (X, t) =
(n)
lim w (X, t)
n→∞ f
≤
Xtn1 −2n2
ρσX γeγXt
λ 1
+ Xtm−n2
2
σ 1−γ
σ(1 − γ)
K
.
(17)
Having established convergence of the perturbation series, we need to compute the higher
order functions gj , j ≥ 1, in order to improve the accuracy of a first order approximation based
on g0 . This task is accomplished next. Specifically, we first define log WT = log Wt + Ht,T for
the wealth dynamics, where
Z
Ht,T :=
t
T
¶
µ
Z T
1 2 2 2n2
n1
ds +
ws σXsn2 dZs .
r + λXs ws − ws σ Xs
2
t
14
Then, we first expand g(X, t) as a power series in γ, we insert it in the equality
·
E
¸
WTγ − 1
eγg(X) W γ − 1
|Ft =
γ
γ
,
(18)
and finally expand also the LHS and the RHS of (18) as a power series in γ. By matching the
resulting coefficient for each power of γ in the resulting power series we obtain the following
characterizations for some of the higher order functions gj (X, t).
Corollary 2. Consider problem (P1). If policies are not binding in the interval [t, T ], we
have:
g0 (X, t) = E [Ht,T |Ft ]
g1 (X, t) =
g2 (X, t) =
g3 (X, t) =
1
Var [Ht,T |Ft ] ,
2
¤´
¤
£ 2
1³ £ 3
|Ft ,
E Ht,T |Ft + 2E [Ht,T |Ft ]3 − 3E [Ht,T |Ft ] E Ht,T
6
¤
¤
£ 2
1³ £ 4
|Ft
E Ht,T |Ft − 6E [Ht,T |Ft ]4 + 12E [Ht,T |Ft ]2 E Ht,T
8
¤2
¤´
£ 2
£ 3
−3E Ht,T
|Ft − 4E [Ht,T |Ft ] E Ht,T
|Ft .
Corollary 2 shows the direct relation that exists between the order of the gj -functions
in the perturbation series and the higher moments of the return Ht,T on optimally invested
wealth. In particular, we see that when the n-order g-function enters is the approximation for
the optimal portfolio policy, the approximation is taking into account the (n + 1)-th moment
of Ht,T . Moreover, the influence of the (n + 1)-th moment on the n-th order approximation
decreases rapidly, i.e. proportionally to 1/(n + 1)!.
To illustrate more concretely the accuracy of the above approximation results we now
compare for an explicit model setting the finite order portfolio policy approximations obtained
with the above procedure and the corresponding exact portfolio policy. We do this for a model
setting corresponding to n1 = 2, n2 = 1, r = 0 in Assumption 2. Exact solutions are obtained
by means of Monte Carlo simulation methods; see for instance7 Cvitanic, Goukasian, and
Zapatero (2003) and Detemple, Garcia, and Rindisbacher (2003).
15
Further, since we perform perturbations exclusively with respect to the risk aversion parameter γ and since our form of VaR constraint is γ-independent, we focus for brevity on an
unconstrained portfolio optimization.
Since for the given model setting we know the log-investor’s portfolio decision in closedform, we use this solution as a control variate. The following parametric model specification
is assumed in the sequel8
dPt
Pt
= 0.05Xt2 dt + 0.25Xt dZt ,
dXt = (0.8 − 0.8Xt )dt + 0.2Xt dZt .
Figure 1 illustrates the findings of our Monte Carlo simulations. Point A in Figure 1 gives
the log-investor’s optimal portfolio policy. The bold straight line in the figure is the firstorder approximation to the underlying portfolio policy. The second-order approximation is
represented by the bold dashed line. Finally, the line with circle marks is the exact portfolio
policy estimated by Monte Carlo simulation. already comes fairly close to the Monte Carlo
simulation. As is apparent from Figure 1 the second order order approximations produces very
accurate results over the given support for the parameter γ. The first order approximation
does not fit the concavity of the portfolio profile as a function of γ, but still produces the
correct sign about the effects obtained for higher/lower risk aversion parameters than in the
log utility case.
2.4
Incorporating Intermediate Consumption
This section extends the previous analysis to incorporate intermediate consumption in the
bank’s optimal wealth dynamics. In this way we can consider the impact of VaR constraints
on the optimal expenditure policy of a bank. Moreover, introducing consumption will allow us
to endogeneize in a later section the risky assets dynamics in the presence of VaR constrained
investors. We interpret intermediate consumption as intermediate cash-outflows caused by the
bank’s expenditure policy. This puts us back to the known portfolio problem where utility is
16
derived from both terminal wealth and intermediate consumption. Assumption 1 is extended
as follows.
Assumption 3. Banks derive utility from both an intermediate consumption process (Ct )0≤t≤T
as well as from terminal wealth WT . They maximize the expected value of the random variable
Z
V (WT , C, T ) =
0
T
e−δs
Wγ − 1
Csγ − 1
ds + e−δT T
,
γ
γ
0 ≤ δ < 1,
(19)
where δ ≥ 0 is the subjective discount rate.
In the presence of intermediate consumption, the wealth dynamics are
dWt = ((r − ct ) + wt λ(Xt ))Wt dt + wt σ(Xt )Wt dZt ,
(20)
where ct = Ct /Wt is the consumption rate. It follows from equation (20) that future current
wealth in the presence of a consumption policy is given by
log WT = log Wt + Ht,T ,
with
Z
Ht,T :=
t
T
µ
¶
Z T
1 2
2
ws λ(Xs ) + r − cs − ws σ(Xs ) ds +
ws σ(Xs ) dZs .
2
t
(21)
(22)
We define gain the relevant VaR limit as a fraction of current wealth, as in (6). Further, recall
that there we assumed a constant portfolio wt over the relevant time horizon (t, t + τ ). In a
similar vain, we define a new VaR constraint where consumption is locked-in at the consumption
rate prevailing at time t. Under Assumption 2 and using the results and the proof of Proposition
2, the Itô-Taylor approximated VaR constraint in the presence of intermediate consumption is
given by
1
1
1
Q(w, c) = r − c + λX n1 w − w2 σ 2 X 2n2 − log(1 − β) + √ vwσX n2 ≤ 0.
2
τ
τ
17
For (c, w), we thus obtain a constrained set C(X) given by
(c, w) ∈ C(X) = {Q(w, c) ≤ 0} ∩ {c > 0}.
Since Q(w, c) is a quadratic function of w and a linear function of c, the set {(c, w) : Q(w, c) =
0} describes a parabola in (w, c)-space. This is illustrated in Figure 2.
The portfolio fractions wb± are obtained as the solutions of the equation Q(c, w) = 0, given
any feasible consumption rate c. Note that the convexity of the constrained set C(X) is ensured
by the fact that ∂ 2 Q(c, w)/∂w2 < 0. Consequently, an increase in volatility due to an increase
of X makes the set C(X) is smaller.
Under Assumption 3, the relevant optimization of a VaR constrained investor is now of the
form
(P2) :
J(W, X, t) =
max
(c,w)∈R(W,X)
E [V (W, C, T ) |Ft ] .
As in the case without consumption, we compute an asymptotic solution for this problem by
exploiting the homogeneity properties of the implied HJB equation.
Proposition 5. Given Assumption 3 and the optimization problem (P2), the first-order approximations of the optimal policies are given by
λ
ρσX ∂g0 (X, t)
,
+ γXtm−n2
2
σ
σ
∂X
1 − γ (g0 (X, t) + log At )
,
At
w(1) (X, t) = (1 + γ)Xtn1 −2n2
c(1) (X, t) =
with At = e−δ(T −t) +
1−e−δ(T −t)
δ
and
1
1 −δ(T −t)
e
E [Ht,T |Ft ] +
g0 (X, t) =
At
At
Z
t
18
T
e−δ(s−t) (E [Ht,s |Ft ] − log As ) ds.
The function E [Ht,T |Ft ] is defined as9
¸
h
i σ2 Z T ·
f
±
n2 2
)
|F
E [Ht,T |Ft ] = E Ht,T
|Ft −
E ll{(wf ,cf )∈C(X)}
(
ŵ
(X,
s)X
t ds,
s
log log /
2 t
Z T
Z
h
i
1 λ2 T h 2(n1 −n2 ) ¯¯ i
f
E Ht,T
|Ft
= r(T − t) −
A−1
ds
+
E Xs
Ft ds,
s
2 σ2 t
t
λ(Xt )
.
ŵ± (X, t) = w± (X, t) −
σ(Xt )2
In Proposition 5 the function E[Ht,T |Ft ] is the sum of two terms. If there were no conf
straints, Ht,T would degenerate to the first term E[Ht,T
|Ft ]. The influence of the second
term in E[Ht,T |Ft ] acts like a utility loss that is caused by the presence of regulatory VaR
constraints. As expected, this second term is related to the regulatory variables β and ν.
Precisely, even when the VaR constraint is not binding today investors anticipate that it could
bind in the future. This reduces the utility that can be expected to be obtained by applying
a given optimal policy and affects the current optimal decision. Thus, the existence of VaR
constraints produces a non trivial anticipative effect on the banks decision. This dynamic
optimal behavior is strongly different from the one implied by a myopic portfolio optimization
as for instance in Danielsson, Shin, and Zigrand (2001).
2.5
Partial Equilibrium: Do VaR Constraints Distort Incentives?
As mentioned in the introduction, Basak and Shapiro (2001) conclude that incentives are
distorted in risk management when a VaR based regulation framework is imposed on the
economy. They find that VaR regulation lets stock market volatility and the risk premium
increase in down markets and decrease in up markets. Refining the assumptions of the Basak
and Shapiro (2001) model, Cuoco, He, and Issaenko (2001) come to a different conclusion
for the partial equilibrium: neither VaR nor any other coherent risk measure give rise to
distortions. Cuoco, He, and Issaenko (2001), however, do not provide a general equilibrium
analysis of VaR incentives and do not consider asset price dynamics with stochastic volatility..
Before embedding our model into general equilibrium, we summarize and briefly discuss the
19
resulting incentives for the bank’s portfolio policies in partial equilibrium. From Proposition 5,
we obtain the following corollary characterizing the solutions of (P2) relative to the portfolio
problem without VaR constraints.
Corollary 3. Consider the partial equilibrium problem (P2) with solutions approximated to
first-order in γ. Before hitting the VaR constraints,
i) banks with γ > 0 increase optimal consumption ct and banks with γ < 0 decrease consumption.
ii) Given ∂g/∂X > 0, banks with γ > 0 increase (decrease) the portfolio fraction invested in
the risky asset if ρ < 0 (ρ > 0). Banks with γ < 0 will decrease (increase) their exposure
in the risky asset.
iii) Given ∂g/∂X < 0, banks with γ > 0 increase (decrease) the portfolio fraction invested in
the risky asset if ρ < 0 (ρ > 0). Banks with γ < 0 will decrease (increase) their exposure
in the risky asset.
If volatility is deterministic and hence the price process follows a geometric Brownian motion,
then the presence of VaR regulation
iii) does not affect portfolio policies.
v) increases consumption ct of banks with γ > 0 and decreases consumption for banks with
γ < 0.
At least two remarks are necessary. First, from the second part of the corollary, we see that
the conclusion of Cuoco, He, and Issaenko (2001) is a peculiarity of their restrictive assumption:
When prices follow a geometric Brownian motion, the VaR constraints have no anticipatory
impact on portfolio strategies. Banks only adjust their portfolio holding when they hit the
VaR constraint. However, VaR constraints influence consumption yet before the constraint
becomes binding.
20
Second, from the first part of Corollary 3, VaR constraints lead either to an increase or to
a decrease in the bank’s exposure to the risky asset. The crucial role is played by, a) the risk
aversion coefficient γ, and b) the correlation ρ between changes in the instantaneous volatility
and the changes in the asset price. The correlation ρ determines the direction of hedging
demand against changes in X (see Proposition 3).
If correlation is negative and γ > 0, VaR has an adverse effect on the riskiness of the banks
portfolio decision. The bank increases its position in the risky asset. Only if correlation is
strictly positive does VaR what the regulator intends it to do: decrease the exposure in the
risky asset. Empirical evidence suggests that stock price movements are negatively correlated.
Such evidence was already reported in Black (1976), Christie (1982), and Schwert (1989), and
is commonly referred to as “leverage effect”. More recent empirical investigations in Anderson,
Benzoni, and Lund (2002) find a correlation10 of ρ = −0.4 for S&P 500 daily returns during
the period from 1/3/1980 to 12/31/1996. If γ < 0, the above effects change signs.
To illustrate the above findings, we provide a graphical analysis of Corollary 3 for a specific
numerical example. We assume a lognormal volatility with mean-reversion. More precisely,
we set λ(X) = λX 2 , σ(X) = σX, and σX (X) = σX X. We first graphically analyze the
optimal portfolio policy when the bank is approaching the VaR constraint. Figure 3 plots
the optimal portfolio strategies for the constrained and unconstrained investor given γ > 0.11
Panel (A) plots the portfolio strategies when the correlation between Xt and the asset return is
positive. For a one year investment horizon, the difference between the two investors are small.
However, in Panel (C) with a five year time horizon, the portfolio fraction in the risky assets
has already been substantially reduced before the bank hits the VaR constraint. Therefore the
mere presence of VaR constraints, although not yet binding, leads to a reduction in the bank’s
exposure to the risky asset. This follows from the fact that the bank has to take into account
the current value of future constraints into today’s investment decision. Similar effects are
shown in Panel (E) for a ten year time horizon. However, if we look at the graphs on the right
of Figure 3, we observe that when ρ is negative the effect of VaR constraints is reversed. As long
as the VaR constraint does not bind, the bank is more exposed in the risky asset than in the
21
absence of VaR constraints. Only when the constraints are binding and the volatility increases
further, the bank will finally end up with a lower exposure than in the unconstrained economy.
Therefore, the effect of VaR regulation strongly depends on the sign of the correlation between
changes in asset returns and changes in asset volatility. As soon as we have ρ < 0, i.e. the
“leverage effect”, the bank increases its exposure in the risky asset. Remarkably, the increase
is more substantial the longer the investment horizon of the bank is. Similar issues follow for
the case γ < 0, but with opposite signs.
As noted in Cuoco, He, and Issaenko (2001), a dynamic VaR setting allows to transform
a VaR limit into an equivalent Expected Shortfall limit, and vice versa (see e.g. Acerbi and
Tasche (2001)). Thus, in our setting with an approximated VaR constraint up to first order,
imposing an Expected Shortfall limit is equivalent to imposing a tighter confidence bound on
the VaR limit. The impact of a tighter confidence bound in the model with stochastic volatility
is plotted in Figure 4 where we use the model specification given above. Again we distinguish
between positive (Panel (A),(C), and (E)) and negative correlation (Panel (B), (D), and (F)).
In Panel (A) and (B) we assume ν = 0.01 and in Panel (C) and (D) we take ν = 0.05. Again,
as in Figure 3, we observe for both confidence bounds an increase in the bank’s exposure
when correlation is negative. In Panel (E) and (F) we plotted the differences between the
portfolio policies under the two confidence levels. Intuitively, one expects portfolio policies to
decrease when moving from ν = 0.05 to ν = 0.01. Surprisingly, this is not quite the case when
correlation is negative. As shown in Panel (F), tightening the confidence level gives the bank
an incentive to increase its exposure at low volatility levels. Only for a volatility larger than
about 50% does a tighter confidence level lead to a decrease in the bank’s risk exposure.
3
General Equilibrium
To fully assess the impact of VaR regulation on the economy, it is not sufficient to analyze
portfolio strategies in a partial equilibrium setting. Indeed, through feedback effects, changes
in portfolio strategies will eventually impact other risk measurement relevant quantities, such
22
as interest rates, volatilities, and risk premia. We therefore extend our model to a general
equilibrium setting. The previous section’s results on partial equilibrium build the basis for
this analysis.
Using perturbation theory, the computation of the general equilibrium amounts to knowing
the optimal policies of the log-investor in a homogeneous economy, avoiding to solve for the
heterogeneous problem ab initio. Admittedly, this methodology does not provide exact results,
but asymptotic approximations. However, comparative statics are exact. Further, as in the
partial equilibrium case, approximations can be improved by considering higher-order terms.
Finally, using perturbation theory the equilibrium expressions can be split into economically
interpretable terms.
3.1
The Exchange Economy
We consider an exchange economy. Our financial market consists of two financial instruments,
an instantaneous risk-free asset and a stock. The risk-free asset is available in zero net supply
and the interest rate rt is determined in equilibrium together with the asset price dynamics.
The risky asset is a contingent claim on aggregate endowment, et , given as
det
= µe (X, t)dt + σe (Xt )dZte ,
et
(23)
with Zte a standard Brownian motion. The state variable Xt follows an Itô diffusion
dXt = µX (Xt )dt + σX (Xt )dZtX .
(24)
The instantaneous correlation between the endowment process and the state variable Xt is
given by E(dZte dZtX ) = ρeX .
In general equilibrium, our model is populated by two banks12 , labelled with indices I and
II, facing problems (P2)I and (P2)II , respectively. The programs (P2)(i) , i = {I, II}, are
equivalent to (P2) with risk aversion index γ replaced by γi . Therefore, the two banks are
23
heterogeneous with respect to their risk aversion only. The variables ωtI and ωtII = 1 − ωtI
denote the cross-sectional distribution of wealth, i.e.
ωti =
Wti
, i = I, II,
WtI + WtII
where Wti is the current wealth of bank i. Compared to the partial equilibrium analysis, the
state of the exchange economy is described by the state vector χt = (Xt , ωtI )> . The cumulative
stock equals aggregate wealth and follows
dPt + et dt
Pt
= α(χ, t)dt + σP 1 (χ, t)dZtX + σP 2 (χ, t)dZte
= α(χ, t)dt + σ(χ, t)> dZt ,
with the drift α(χ, t) and the vector σ(χ, t) determined in equilibrium.
Definition 2. We call (α(χ, t), σ(χ, t), r(χ, t), wtI , wtII , cIt , cII
t ) an asymptotic pure exchange
equilibrium if
a) individual portfolio rules wtI , wtII are optimal to first order, i.e., they satisfy (14).
b) the financial market clears,
wtI ωtI + wtII ωtII = 1 + O(γ 2 ).
(25)
b) the market of consumption goods clears,
II
2
cIt WtI + cII
t Wt = et + O(γ ).
3.2
(26)
Equilibrium Policies
We derive first-order asymptotic expressions for the individual policies in a heterogeneousagent economy, where all agents solve problem (P2) with intermediate consumption. Two
cases are considered. In the first benchmark case the economy is not constrained. In the sec24
ond case, the regulator constrains banks with a VaR limit as outlined in the previous sections.
The functional form of the portfolio and consumption policies are the one obtained in partial
equilibrium. However, the function g0 in Proposition 5 is replaced by some corresponding
equilibrium function g0e , which now depends on the state variables, Xt and ωtI , and the preference parameters γI and γII . Since g0e depends on equilibrium quantities, it is endogenously
determined. Expanding g0e with respect to the risk aversion parameters, we obtain

g0e (χ, γI , γII , t) = g0,0 (χ, t) + 
>
γI
γII
2
 g0,1 (χ, t) + O(γI2 , γII
).
2 ). The funcFor notational brevity we write g0e (χ, γI , γII , t) = g0e (X, t), and O(γ 2 ) for O(γI2 , γII
tion g0,0 is uniquely determined by the log-investor’s value function in an economy populated
by log-investors solely.
With the above specifications the portfolio policies in the general equilibrium for investor
j, j = {I, II} up to first order are
wtj∗ = (1 + γj )
cj∗
=
t
λ∗ (χ, t)
+ γj ηt∗ + O(γ 2 ),
kσ ∗ (χ, t)k2
1
− γj ξt∗ + O(γ 2 ),
At
(27)
(28)
where
ξt∗ =
ηt∗ =
∗ (χ, t) + log A
g0e
t
,
At

>
∗
∗
σ (χ, t)
1
 PX
 ∂g0e (χ, t) ,
kσ ∗ (χ, t)k2
∂χ
σP∗ ω (χ, t)
(29)
(30)
with σP∗ X (χ, t) and σP∗ ω (χ, t) the covariances between stock price changes and the state variables X and ω I . In the above equations, we use ‘∗’ as a placeholder for either ‘f ’ (unconstrained
case) or ‘c’ (constrained case). Before summarizing first properties of the unconstrained and
constrained economy, we specify the function g0,0 , which can be calculated explicitly, in the
25
lemma below.
Lemma 2. With an endowment and state variable process given in (23) and (24), the function
∗ (χ, t) up to order O(γ) is given by
g0e
∗
g0e
(χ, t) = g0,0 (X, t) + O(γ)
Z T
£ e
¤
£ e
¤¢
e−δ(T −t) ¡
1
e−δ(s−t) E Ht,s
|Ft ds + O(γ),
=
log At + E Ht,T |Ft +
At
At t
with
E
£
e
Ht,T
|Ft
¤
Z
=
t
T
·
¸
1
2
E µe (X, s) − σe (X, s) |Ft ds .
2
It is important to note that the function g0,0 is the same for the economy with and without
VaR-constraints in equilibrium. This is due to the market clearing conditions in a homogeneous
log-economy: Investment in the risky asset has be equal to 1 irrespective of any VaR-constraint.
Moreover, in the log-economy, g0,0 is independent of cross-sectional wealth, since the portfolio
fractions in the homogenous log-economy always move in the same direction. As a consequence, in a constrained economy banks will not change their portfolio policies up to first
order in γ due to the possibility of being constrained in the future. This differs from partial
equilibrium analysis, where banks adjust their asymptotic portfolio policies in the presence of
VaR constraints.
As ξt∗ will not be affected by the presence of VaR regulation, we write ξt∗ = ξt and also for
∗ . Also, note that ξ is a function of X and t only, but not of ω I , contrary to η , which depends
g0e
t
t
t
on cross-sectional wealth. However, since g0e (X, t) becomes independent of ω, equation (30)
simplifies to
ηt∗ =
1
∂g0e (χ, t)
σP∗ X (χ, t)
.
∗
2
kσ (χ, t)k
∂X
(31)
The individual optimal policies are given in equations (27) and (28). As follows from the next
proposition, we only need to calculate g0e (X, t) up to order O(γ) to obtain second-order asymp26
totic equilibrium quantities. We define by ∆∗ a weighted aggregated risk aversion coefficient
∆∗ = γI ωtI∗ + γII (1 − ωtI∗ ).
The next proposition summarizes the properties of the unconstrained equilibrium.
Proposition 6. With an endowment and state variable process given in (23) and (24) and no
VaR-constraints,
i) the equilibrium interest rate is given by
³
´
r(χf , t) = α(χf , t) − kσ(χf , t)k2 1 − (1 + ηtf )∆f + O(γ 2 ).
ii) the cross-sectional wealth dynamics of investor I is given by
dωtIf
³
´³
³
´
´
= (γI − γII ) ωtIf 1 − ωtIf ξt dt + 1 + ηtf σe (Xt )dZte + O(γ 2 ).
iii) drift and volatility of the cumulative price process in (25) are given by
µ
αf (χf , t) = δ + µe (Xt ) + (∂t At − 1)ξt ∆f + At ∆f

σ(χf , t) = 
σPf 1 (χf , t)
σPf 2 (χf , t)


=
∂ξt
At ∆f ∂X
σX (Xt )
t
∂ξt
∂ξt
+ σeX
∂t
∂Xt

¶
+ O(γ 2 ),
 + O(γ 2 ).
σe (Xt )
It follows from this proposition that as soon as ξ is fixed, which corresponds to choosing
a particular process for the endowment process, the risky asset and cross-sectional wealth
dynamics are explicitly given. We then can explicitly calculate dξ. The cross-sectional wealth
dynamics are non-linear in general (see e.g. Trojani and Vanini (2001)). Further, since dωtI is of
order O(γ), the drift and the variance of the equilibrium price process only involve infinitesimal
generators of X, but not of ωtIf .
We next turn to the characterization of the constrained equilibrium. In order to obtain
tractable expressions, we use the fact that the VaR horizon, τ , is a short time interval, typically
27
a day. Therefore, we expand the resulting expressions in τ and consider terms only up to second
order. We furthermore assume without loss of generality that bank II is less risk averse than
bank I.
Proposition 7. Given an endowment and state variable process given in (23) and (24). In
an economy with VaR-constraints where bank II is constrained and I not,
i) the equilibrium interest rate is given by
µ
¶
v(1 − ωtIc )
1 √
r(χ , t) = α(χ , t) − kσ(χ , t)k + kσ(χ , t)k p
+ h γI ; √ , τ , τ + O(γ 2 , τ 2 ),
τ
ωtIc B
c
c
c
2
c
√
with B = ωtIc v 2 + 2 log(1 − β)(ωtIc − 2) and h(γI ; √1τ , τ , τ ) given in the appendix.
ii) the cross-sectional wealth dynamics of bank I is given by
dωtIc =
¡ ωc c
¢
c
ωc c
ωc c
e
µ0 (χ , t) + µωc
γ (χ , t) + µτ (χ , t) dt + σ0 (χ , t)dZt
¡
¢>
+ στωc (χc , t) + σγωc (χc , t) dZt + O(γ 2 , τ 2 ),
c
ωc c
where the functions (µωc
· (χ , t), σ· (χ , t)) are functions of the state variables and the
regulatory parameters.
iii) drift and volatility of the cumulative price process in (25) are given by
µ
µ
¶
¶
∂ξt /∂X
ωc c
ωc c
α (χ , t) = (γI − γII )At ξt µ0 (χ , t) + ρeX σX (Xt )
+ σe (Xt ) σ0 (χ , t)
ξt
c
c
+αf (χc , t) + O(γ 2 , τ 2 ),


f
c , t)
(χ
σ
P1
 + O(γ 2 , τ 2 ),
σ(χc , t) = 
σe (Xt ) + (γI − γII )ξt σ0ωc (χc , t)
where αf (χc , t) and σPf 1 (χc , t) are given in Proposition 6 with ωtIf replaced by ωtIc . The
c
ωc c
functions µωc
0 (χ , t) and σ0 (χ , t) are given in (A.41) and (A.42).
By inspection of Propositions 6 and 7, the cross-sectional dynamics in the constrained
equilibrium do not vanish in O(γ) as in the unconstrained economy. However, for determining
28
the equilibrium quantities in an O(γ 2 )-asymptotic economy, terms up to order O(γ) of the
cross-sectional wealth dynamics do matter. Therefore, volatility and drift of the asset price
process do change in the constrained equilibrium. However, the equilibrium interest rate
changes irrespective of the cross-sectional dynamics’ order in γ.
Before discussing different model specification, it is instructive to consider a log-economy.
Since in our equilibrium setting with only two banks an equilibrium cannot exist with both
banks constrained, we assume that bank I is not regulated. In this economy, the only heterogeneity stems from different initial wealth levels of the investors. The corollary below
summarizes the equilibrium quantities for a log-economy.
Corollary 4. Given an endowment and state variable process given in (23) and (24) with bank
II VaR-constrained and bank I unregulated. When γI = γII = 0,
i) the equilibrium interest rate is given by
µ
¶
v(1 − ωtIc )
1 √
r (χ, t) = r (χ, t) + kσ(χ, t)k p
+ h 0; √ , τ , τ + O(τ 2 ),
τ
ωtIc B
c
f
rf (χ, t) = α(χ, t) − kσ(χ, t)k2 .
ii) drift and volatility of the cumulative price process in (25) are given by
αc (χ, t) = αf (χ, t) = δ + µe (Xt ),
σ c (χ, t) = σ f (χ, t) = σe (Xt ).
In a log-economy, drift and volatility of the equilibrium asset price process remain unchanged. Only interest rates do change as well as the asymptotic volatility and drift of the
cross-sectional wealth dynamics. Hence, regulation induces a change in wealth distribution
among financial institutions even in a homogeneous log-economy. This is remarkable, since
log-investors have zero dynamic hedging demand with respect to the state variables and therefore interest rates are the same independent of the existence of a stochastic opportunity set.
Second, our choice of VaR limit (6) implies a wealth independent approximate VaR constraint.
29
This shows that an regulatory standard may well have an equilibrium impact on economic
variables which are not part of the definition of the standard.
3.3
General Equilibrium: Do VaR Constraints Distort the Economy?
We next explore the effect of different model specifications M of the endowment process on
the general equilibrium. We choose the models:



M0 : µe (X) = µe , σe (X) = σe Xt , γI = γII = 0;



M:
M1 : µe (X) = µe , σe (X) = σe X;




 M2 : µe (X) = µe X , σe (X) = σe X.
Furthermore, we assume that the state variable X follows a mean-reverting geometric Brownian
motion. Alternatively, we could assume a square-root or Gaussian process for X. However,
the results do not change qualitatively.
Our main focus is the VaR impact on equilibrium interest rates, volatility, portfolio fractions
and risk premia. With these model specifications at hand, we conduct a graphical analysis
(Figures 5-8) of an economy, where bank II is subject to VaR regulation, whereas bank I is
not constrained.
First, we discuss model M0 , i.e. the log-economy. Corollary 4 states that in a log-economy
VaR regulation has no effect on the asset price dynamics under the physical probability measure. Drift and volatility are the same whether bank II is constrained or unconstrained. However, as soon as bank II becomes constrained, interest rates change even in a pure log-economy.
Only in the limit,
lim = αc (χ, t) − kσ(χ, t)k2
ω→1
1 − γI η t
,
1 + γI
interest rates remain unchanged compared to the unconstrained log-economy.
Figure 5 plots the asymptotic portfolio fractions and the equilibrium interest rates as well
as the asymptotic drift and volatility of the cross-sectional wealth process. In the first panel,
left to point A, the portfolio fractions equal one, i.e. w = 1 for both banks. To the right of
30
point A, the regulated bank II becomes constrained and therefore has to decrease its exposure
(dashed line). Part of its wealth are now shifted into the money market. This influences
the equilibrium interest rates, which have to decrease due to the increased demand for bonds
from bank II. The bold straight line in the upper right panel shows the equilibrium interest
rate. When point A is reached, the interest rates become lower than in the corresponding
unconstrained economy (dotted line). Lower interest rates make an investment in the risky
asset more attractive for the unregulated bank, which, to the right of point A, increases its
exposure (straight bold line in the upper left panel). The changes in portfolio fractions will
induce changes in the cross-sectional wealth dynamics in leading order in γ. This means that
regulation causes changes in the wealth distribution regardless of the risk-appetite of regulated
and non-regulated institutions. In the lower panels, we plot the drift component µω0c (χ, t) and
ω (χ, t) of cross-sectional wealth dynamics. First, we note that due
the volatility component σ0c
to regulation, cross-sectional wealth dynamics become stochastic in O(γ). Second, we note
that the drift of cross-sectional wealth becomes positive in O(γ). Therefore, regulation favors
unregulated institutions as it eventually leads to a shift of cross-sectional wealth from the
regulated bank to the unregulated bank.
In Figure 6 we discuss model M1 assuming a positive correlation ρeX . As soon has bank
II hits the VaR constraint at point A, it has to decrease its portfolio fraction invested in the
risky asset. The dotted line through point A plots the VaR constraint. This VaR constrained
is determined by the interest rate, drift and volatility of the asset price process in an unconstrained economy. As soon as bank II hits point A, these equilibrium quantities change to
their corresponding values in the constrained economy. Therefore, the VaR constraint will
change at point A to the lower dotted line. As equilibrium quantities change in a non-smooth
manner, there will be a downward jump in the portfolio fraction at point A. The increased
demand for bonds leads to a decrease in interest rates (upper-right panel). Thus, beyond point
A, the riskless interest rate starts to decline. At the same time, we observe an upward jump
in the asset’s drift rate and an increase its volatility. Thus, in case of positive correlation ρeX ,
regulation induces an upward jump in equilibrium volatility. Only when X increases further,
31
volatility will eventually be lower than in the unconstrained economy.
Since interest rates, drift and volatility have to change in such a way that bank I increases
its demand for stocks (point B in Figure 6), the overall effect is an increase in the Sharpe
Ratio. Figure 7 plots the Sharpe Ratio for the parameter used in Figure 6. At point A the
Sharpe Ratio jumps to a higher level and obtains a steeper slope in X than the Sharpe Ratio
of the unconstrained economy. Thus, the presence of regulated and non-regulated financial
institutions may help to explain in part the stylized fact of relatively high Sharpe Ratios
together with low interest rates, which standard financial models fail to explain.
In Figure 8 we explore model M2 with negative correlation ρeX . Again, we assume that
bank I is not regulated. Since we assume correlation to be negative, the portfolio fraction
invested in the risky asset is lower for bank II than for bank I in the unconstrained economy.
When bank II becomes constrained at point A, it has to decrease its portfolio fraction. This
leads to a decrease in interest rates at point A in the upper-right panel. Compared to model
M1 with negative correlation, it turns out that, when bank II becomes constrained, both drift
and volatility of the asset price process jump downwards and remain at a lower level compared
to the unconstrained economy. Eventually however, the Sharpe Ratio will increase and bank I
will increase its exposure in the risky asset at point B as shown in the first panel of Figure 8.
We have now analyzed different models under different parameter constellations. It follows
that to fully understand the impact of VaR regulation on equilibrium quantities, such an
analysis has to be carefully considered at least along three dimensions:
1. Regulatory control variables: β, ν, τ .
2. Market factors: X, ρeX .
3. Market participants characteristics: γ, ω, T .
All the above variables have different and mostly ambiguous impacts on the effectiveness of
VaR regulation. In contrast to the analysis in partial equilibrium, the effect of VaR regulation
is hardly predictable. The reason for this lies in the fact that risk regulation is no longer
32
exogenously given, but enters as an endogenous constraint into the optimization problem of
the bank. The VaR constraint is calculated using interest rate, asset price drift and volatility
as inputs. These quantities are determined endogenously in equilibrium and, thus, complicate
the equilibrium analysis of VaR constraints by introducing highly non-linear effects and jumps.
There are nevertheless some important conclusions to be drawn from the above analysis:
• Regulation leads to a change in cross-sectional wealth, which is independent of risk
aversion. Wealth is shifted from the regulated to the unregulated institutions.
• For unregulated institutions, an investment in the risky asset becomes more attractive,
since regulation leads to an increase in the Sharpe Ratio.
• When the regulated bank becomes constrained, equilibrium interest rates, asset price
drift and volatility can jump.
• When ρeX is positive and bank II becomes constrained, the Sharpe ratio seems to be
mostly augmented by an increase in the drift rate. Whereas with negative correlation,
the Sharpe ratio tends to be augmented by a decrease in volatility.
In summary, the exact effects of VaR regulation are difficult to establish in a fairly realistic
model. Considering the costs, which the implementation of regulatory standards are causing
to the financial industry, one would expect that regulation leads to more decisive statements.
However, present and planned regulation fails to provide such a statement.
As a final remark, our framework can also be applied to analyze the case of a large bank
with VaR limited traders that can move the market.
4
Conclusion
We analyze the implications of VaR based regulation on the investment policies of utility maximizing investors in a continuous-time setting. We extend the existing literature by allowing
33
for a stochastic opportunity set and intermediate consumption expenditures. In partial equilibrium we showed that VaR regulation can pronounce the bank’s exposure in the risky asset.
This distortion of incentive is however not related to the lack of coherence of VaR. It instead
depends on the correlation between asset price changes and changes in volatility and the degree of risk aversion. If correlation is negative, for a bank which is less risk-averse then the
log-investor, the presence of VaR limits increases the risky asset exposure. Furthermore, it is
impossible to remove any incentive distortions by replacing VaR with a coherent risk measure
such as Expected Shortfall: Market factors and preferences of the investors, which are both
linked to the VaR constraint, induce the distortions.
Extending our analysis to the general equilibrium, we explore the effect of VaR regulation
on equilibrium interest rates, volatility, portfolio fractions, and we analyze the effect on the
banks’ portfolio policies. Applying the general results to some well-known models, we find
that impacts of VaR regulation strongly depend on both the model used and the parameter
values in the economy. Indeed, to fully assess the effectiveness of VaR regulation, the analysis
has to be done along three dimensions, a) the market participants, b) the market factors, and
c) the regulatory control variables. Conclusions remain mixed, however, we find a tendency
for lower interest rates and an increase in the Sharpe Ratio. Furthermore, regulation leads
to changes in cross-sectional wealth, which are independent of risk aversion. Therefore, regulation distorts competition among regulated banks, and between regulated and unregulated
financial institutions. As the costs for the financial industry to implement regulatory standards are considerable, one would expect that the effect of regulation can be summarized in a
clear statement. However, within an adequate model, such a statement cannot be made. One
reason for this lies in the endogenization of regulation by using VaR as a regulatory control
mechanism. Such an endogenization cannot be circumvented by using Expected Shortfall or
other coherent risk measures, which depend on equilibrium quantities such as interest rates
and volatilities.
34
w
0.2
0.5
0.8
w
0.2
0.5
0.8
w
0.2
0.5
0.8
1 day 10 days 1 day 10 days
X=1
X=3
M = 1%
0.000
0.001
0.002
0.189
0.000
0.005
0.005
0.525
0.000
0.009
0.009
0.924
M = 5%
0.000
0.003
0.000
0.038
0.000
0.001
0.001
0.105
0.000
0.002
0.002
0.185
M = 10%
0.000
0.000
0.000
0.019
0.000
0.000
0.001
0.053
0.000
0.001
0.001
0.092
Table 1: First-Order Approximation. The table displays the upper bound for the probabilities
given in Proposition 1. Reported numbers are percentage numbers. We consider the following
model specification: dX = (θ − κXt )dt + σX XdZtX , r(Xt ) = r, λ(Xt ) = λX 2 , and σ(X) = σX.
To calculate the values for different M , X, w and time-horizons, we assumed θ = 0.2, κ = 0.2,
σX = 0.15, λ = 0.05, and σ = 0.2. The unconditional mean is set to θ/κ = 1.
35
0.75
Monte Carlo Simulation
First−Order Approximation
Second−Order Approximation
0.7
portfolio fraction
0.65
0.6
A
0.55
0.5
0.45
0.4
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
risk aversion
Figure 1: Comparing perturbation approach with Monte Carlo simulation. We assume α(Xt ) =
0.05Xt2 , σ(Xt ) = 0.25Xt , θ = κ = 0.8, σX = 0.2, T = 1.
36
consumption rate c
feasible set
{Q(c,w)≤ 0} ∩ { c≥ 0 }
+
(wb, c = 0)
(w−, c = 0)
b
portfolio fraction w
Figure 2: VaR constraint set parameterized by consumption and portfolio value.
37
(B)
(A)
portfolio fraction w
portfolio fraction w
0.36
0.365
0.36
0.355
0.355
0.35
0.345
0.35
0.34
0.2
0.4
0.6
0.8
(C)
1
0.2
σ(X)
portfolio fraction w
portfolio fraction w
0.37
σ(X)
1
0.8
σ(X)
0.8
σ(X)
0.34
0.33
0.32
0.2
0.4
0.6
0.8
(E)
1
0.2
σ(X)
0.4
0.6
(F)
1
0.4
portfolio fraction w
0.45
portfolio fraction w
0.8
0.35
0.38
0.425
0.4
0.375
0.35
0.6
(D)
0.39
0.36
0.4
0.2
0.4
0.6
0.8
1
0.35
0.3
0.25
0.2
σ(X)
0.4
0.6
1
Figure 3: Portfolio policy under VaR and influence of investment horizon. Graphs on the left
assume a positive correlation of ρ = 0.4. Graphs on the right assume ρ = −0.4. (A) and (B)
assume an investment horizon of T = 1 year. The next two graphs assume T = 5, and the
final two graphs assume T = 10. The solid bold line represents the portfolio policy of the VaR
constrained bank. At the circled point, the bank runs into the VaR constrained represented
by the dotted line. The dashed line represents the optimal policy in case of no constraints.
The parameters were chosen as follows: γ = 0.5, ν = 0.01, β = 0.05, τ = 10/250, r = 0.05,
λ = 0.03, σ = 0.32, κX = θX = 0.2, σX = 0.6. The calculations were performed using standard
Monte Carlo methods.
38
(B)
(A)
0.49
portfolio fraction w
portfolio fraction w
0.54
0.53
0.52
0.51
0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(C)
0.48
0.47
0.46
0.45
0.1
0.8
σ(X)
0.2
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.7
(D)
0.8
σ(X)
portfolio fraction w
portfolio fraction w
0.54
0.53
0.52
0.51
0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
−0.05
−0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.44
0.2
0.3
0.4
(F)
0
0.1
0.46
0.1
σ(X)
portfolio difference
portfolio difference
(E)
0.48
0.8
0
−0.05
−0.1
0.1
σ(X)
0.8
σ(X)
0.2
0.3
0.4
0.8
σ(X)
Figure 4: Portfolio policy under VaR and the influence of confidence level. Graphs on the left
assume ρ = 0.4 and ρ = −0.4 on the right side. (A) and (B) assume a confidence bound of
ν = 0.01. (C) and (D) assume ν = 0.05. (E) and (D) show the differences in optimal policies
when the confidence level is tightened, i.e. we plot the differences (C)-(A) and (D)-(B). We
assumed the same parameters as in Figure 3. The time horizon is set to T = 5 years.
39
portfolio fraction
interest rate
1.6
1.4
A
0.1
A
r(χ,t)
w(χ,t)
1.2
1
0.8
0.05
0.6
0.4
0.5
1
1.5
state variable X
0
0.5
2
1
µω (χ,t)
1.5
state variable X
2
σω (χ,t)
0c
0c
0.03
0.1
0.025
0.08
0.02
σω (χ,t)
0c
µω
(χ,t)
0c
0.06
0.015
0.04
0.01
0.02
0.005
A
A
0
0
0.5
1
1.5
state variable X
2
0.5
1
1.5
state variable X
2
Figure 5: Impact of VaR regulation in a log-economy M0 . We assumed the following parameter
values: σe = 0.2, µe = 0.1, σX = 1, θ = κ = 0.1, δ = 0.05, T = 5, τ = 1/250, β = 2.5%, v =
5%, ω = 0.5, γI = γII = 0. Bold lines represent quantities resulting from the constrained
economy. In the panels for the interest rate, drift and volatility of the cross-sectional wealth
dynamics, the dotted lines to the right of point A represent the corresponding quantities in
the unconstrained economy.
40
portfolio fraction
1.6
interest rate
A
1.4
0.1
1
r(χ,t)
w(χ,t)
1.2
A
0.8
0.05
0.6
0.4
0.2
0.5
B
1
state variable X
0
0.5
1.5
drift
1.5
volatility
0.1
0.26
0.095
0.24
0.22
0.09
A
σ(χ,t)
α(χ,t)
1
state variable X
0.085
0.2
A
0.18
0.08
0.16
0.075
0.14
0.5
1
state variable X
1.5
0.5
1
state variable X
1.5
Figure 6: Impact of VaR regulation for M1 with positive correlation. We assumed det /et =
µe dt + σe Xt dZte and the following values: σe = 0.2, µe = 0.1, σX = 1, θ = κ = 0.1, δ = 0.1, T =
2, τ = 1/250, β = 5%, v = 1%, ω = 0.5, ρeX = 0.4, γI = −0.1, γII = 0.4. Bold lines represent
quantities resulting from the constrained economy. In the panels for the interest rate, drift
and volatility of the asset price process, the dotted lines to the right of point A represent the
corresponding quantities in the unconstrained economy.
41
Sharpe Ratio
0.7
0.6
0.5
φ(χ,t)
0.4
0.3
0.2
A
0.1
0
−0.1
0.5
1
1.5
2
state variable X
Figure 7: Impact of VaR regulation on Sharpe Ratio for M1 with positive correlation. We
assumed the same parameter values as for Figure 6. The bold line is the Sharpe Ratio resulting
from the constrained economy. The dotted line to the right of point A represents the Sharpe
Ratio in the unconstrained economy.
42
portfolio fraction
interest rate
2
0.07
0.06
1.6
0.05
1.4
0.04
1.2
r(χ,t)
w(χ,t)
1.8
B
0.03
1
0.02
A
0.8
0.6
0.5
1
state variable X
0.01
0
0.5
1.5
drift
0.28
0.12
0.26
0.11
0.24
1.5
0.22
A
σ(χ,t)
α(χ,t)
0.1
A
0.2
0.08
0.18
0.07
0.16
0.06
0.05
0.5
1
state variable X
volatility
0.13
0.09
A
0.14
1
state variable X
1.5
0.5
1
state variable X
1.5
Figure 8: Impact of VaR regulation for M2 with negative correlation. We assumed det /et =
µe Xt dt + σe Xt dZte and the following values: σe = 0.2, µe = 0.1, σX = 1, θ = κ = 0.1, δ =
0.1, T = 2, τ = 1/250, β = 2.5%, v = 1%, ω = 0.5, ρeX = −0.4, γI = −0.1, γII = 0.4. Bold lines
represent quantities resulting from the constrained economy. In the panels for the interest rate,
drift and volatility of the asset price process, the dotted lines to the right of point A represent
the corresponding quantities in the unconstrained economy.
43
Appendix
Proof of Proposition 1
With portfolio weights fixed at time t = 0, the wealth dynamics under Assumption 2 reads
Z
log Wt = log W0 +
0
t
1
(r(Xs ) + w0 µ(Xs ) − w02 σ 2 (Xs ))ds + w0
2
Z
t
σ(Xs )dZs .
0
Performing an Itô-Taylor expansion in X, we obtain log W = log W (1) + R with remainder
R =
Z tZ s
Z tZ s
¡
¢
w0 σ(X0 )
dZu ds +
L0 h1 (Xu )du + L1 h1 (Xu )dZuX ds
0
0
0
0
Z tZ s
¡
¢
+w0
L0 h2 (Xu )du + L1 h2 (Xu )dZuX dZs ,
0
(A.32)
0
where h1 (X) = r(X) + w0 λ(X) + 12 w02 σ(X)2 , h2 (X) = σ(X), and
L0
= µX (X)
∂
1 2
∂2
∂
+ σX
(X)
, L1 = σX (X)
.
∂X
2
∂X 2
∂X
Using Markov’s inequality we get the bound
¯
³¯
´
¯
¯
(1)
P ¯log Wt − log Wt ¯ ≥ M
≤
¯i
1 h¯¯
1
¯
(1)
E ¯log Wt − log Wt ¯ =
E [|R|] .
M
M
Since the first moment of a multiple Itô integral vanishes, if it has at least one integration with respect
to a component of the Wiener process, E [|R|] follows as claimed.
Proof of Proposition 2
From Proposition 1,
¯ ¢
¡
P Wt+τ − Wt ≤ L¯ Ft
!
Ã
¢
¡
¯
log(1 − β) − r(Xt ) + wt λ(Xt ) + 12 wt2 σ(X)2 τ
¯
+ R Ft
= P Zt+τ − Zt ≤
wt σ(Xt )
³
´
(1)
≈ P Wt+τ − Wt ≤ L| Ft
Ã
¡
¢ !
log(1 − β) − r(Xt ) + wt λ(Xt ) + 21 wt2 σ(X)2 τ
√
= N
,
wt σ(Xt ) τ
44
¤
where β = L/Wt and N (·) the cumulative normal distribution function. The approximated VaR at the
ν- confidence level then reads
ν,w
dt
VaR
³
√ ´
2
1 2
= Wt 1 − e(r(Xt )+wt λ(Xt )− 2 wt σ(Xt ) )τ +vwt σ(Xt ) τ ,
(A.33)
which is equivalent to
¶
µ
√
1 2
2
log(1 − β) − r(Xt ) + wt λ(Xt ) − wt σ(Xt ) τ − vwt σ(Xt ) τ ≤ 0.
2
But this inequality is equivalent to the upper and lower bound wb± stated in the proposition.
¤
Proof of Proposition 4
We calculate the value function for a log-investor as
Jlog (W, X, t) =
E [log WT |Ft ] ,
¶
Z Tµ
¤
1 £ 2 2
= log Wt +
r + E [wlog λ(Xs ) |Ft ] − E wlog σ (Xs ) |Ft ds ,
2
t
with


wf


 log
wlog =




(A.34)
f
if wb− < wlog
< wb+ ,
wb+
f
if wlog
≥ wb+ ,
wb−
f
if wlog
≤ wb− ,
where
f
wlog
= X n1 −2n2
λ
,
σ2
is the log-investor’s portfolio policy in the unconstrained case. Since
eγ(g0 (X,t)+γg1 (X,t)) Wtγ − 1
= log Wt + g0 (X, t),
γ→0
γ
Jlog (W, X, t) = lim
(A.35)
holds, equating (A.34) and (A.35) g0f (X, t) follows. Accounting for the presence of constraints, we have
Z
g0 (X, t)
= r(T − t) + λ
1
− σ2
2
Z
t
·³
T
E
t
T
h
f
E wlog
(Xs ) + ll{|wf
i
ŵb± (Xs )Xsn1 |Ft ds
¸
´2
±
2n2
X
|F
± ŵ (Xs )
t ds.
s
|>|w |} b
±
log |>|wb |}
f
wlog
(Xs ) + ll{|wf
log
45
b
Simplifying the above term, we arrive at the decomposition for g0 (X, t) as given in the theorem.
¤
Proof of Corollary 1
Plugging the bound KX 2j(n1 −n2 ) into equation (16), we can use the Euler gamma function Γ(a) and
the incomplete Gamma function, Γ(b, a) to obtain
(n)
wf (X, t)
= Xtn1 −2n2
λ 1 − γ n+1
σ2 1 − γ
ρσX
eX
γKXtm−n2
+
σ
2(n1 −n2 )
γ
Γ(n, X 2(n1 −n2 ) γ) − γ n eX
(1 − γ)Γ(n)
2(n1 −n2 )
Γ(n, X 2(n1 −n2 ) )
.
Since limb→∞ Γ(b, a)/Γ(b) = 1 for finite a, the claim follows.
Proof of Proposition 5
The HJB equation for the control problem (P2) is
(
0
=
1
max Jt + µX (X)JX + σX (X)2 JXX + W (r − c + wλ(X)) JW
w
2
1
+wW ρσ(X)σX (X)JW X + w2 W 2 σ(X)2 JW W
2
µ
µ
¶
¶)
√
1 2
2
−φ log(1 − β) − r − c + wλ(X) − w σ(X) τ − wσ(X) τ v
2
=: max (−φ1 Q(w, c) − φ2 c + GJ) .
w,c
Due to the homogeneity properties of the utility function, the solution is of the form
J(W, X, t) =
´
At ³ γg(x,t) γ
e
W −1 .
γ
The rirst-order conditions and slackness imply
wf
cf
σX JW X
λJW
− ρX m−n2
σ 2 W JW W
σW JW W
λ
γ
ρσX ∂g(X, t)
= Xtn1 −2n2 2
+
Xtm−n2
,
σ (1 − γ) 1 − γ
σ
∂X
1
´ γ−1
³
1
= (JW W ) γ−1 = At eγg(X,t)
.
=
−X n1 −2n2
46
(A.36)
As we are only interested in the first-order approximation, we assume g(X, t) = g0 (X, t) + γg1 (X, t) +
O(γ 2 ). From (A.36) the log-investor’s value function reads
J(W, X, t) = At (log(Wt ) + g0 (X, t)) .
Expanding the optimal portfolio weight and optimal consumption rate up to first order we obtain
(1)
(1)
wf (X, t) and cf (X, t) as claimed. To obtain the function g0 (X, t), we recall the wealth dynamics
with consumption which is given in equation (21). Then, the J function reads
J(W, X, t)
= e−δ(T −t) (log Wt + E [Ht,T |Ft ])
Z T
+
e−δ(s−t) (log Wt + E [Ht,s |Ft ] + E [log cs |Ft ]) ds.
t
Equating the last two expressions for the value function we get
1 − e−δ(T −t)
,
δ
At
=
e−δ(T −t) +
g0 (X, t)
=
1 −δ(T −t)
1
e
E [Ht,T |Ft ] +
At
At
Z
T
e−δ(s−t) (E [Ht,s |Ft ] + E [log cs |Ft ]) ds.
t
We still need to determine the function Ht,T . Its general form is given in equation (22). Similar as in
Proposition 4, the structure of w± (X, t) leads to the decomposition of Ht,T as given in the theorem. ¤
Proof of Corollary 3
To prove the increase in consumption ct , we have to show that gf (X, t) > g0 (X, t). For the nonh
i
f
degenerated case this follows from E Ht,T
|Ft > E [Ht,T |Ft ]. To prove the impact on the portfolio
fractions, we have to show
∂gf (X, t)
∂g0 (X, t)
<
.
∂X
∂X
(A.37)
From the definition of E [Ht,T |Ft ] in Proposition 5, the first derivative of the Heaviside function with
respect to X is positive, the first derivative of the function ŵt± is negative and the derivative of the
volatility function is positive by assumption. Then (A.37) follows.
47
¤
Proof of Lemma 2
The proof of Proposition 5 implies that the function g0e (χ, t) must be of the form
1 −δ(T −t)
g0e (χ, t) =
e
E [Ht,T |Ft ] +
At
with
"Z
E [Ht,T |Ft ] = E
t
T
RT
t
e−δ(s−t)
(E [Ht,s |Ft ] + E [log cs |Ft ]) ds,
At
#
µ
¶
1 2 2
rlog + wlog (αlog − rlog ) − clog (s) − wlog σlog ds |Ft .
2
As g0e (χ, t) is determined in the homogenous log-economy, the above equation can be simplified considerably. First, we note that in the log-economy, market clearing implies wlog = 1. Further, clog = 1/At
and, as γI = γII = 0,
αlog = δ + µe (X, t),
σlog = σe (Xt ).
Inserting the above expressions in g0e (χ, t) proves the claim.
¤
Proof of Proposition 6
To obtain the equilibrium interest rate rf (χ, t) we start with the market clearing condition
µ
ωtI
α(χ, t) − r(χ, t)
(1 + γI ) + γI ηt
kσ(χ, t)k2
¶
µ
+ (1 −
ωtI )
α(χ, t) − r(χ, t)
(1 + γII ) + γII ηt
kσ(χ, t)k2
¶
= 1.
Solving for r(χ, t) and expanding in γ gives the equilibrium interest rate. To determine dωtI , recall that
ωtI = WtI /Pt .
Applying Itô’s Lemma to cross-sectional wealth, inserting the wealth differentials and
dropping the O(γ 2 ) terms, we obtain the expression for dωtIf up to second order in γ. To derive the
price process, we consider
d(Pt /et ) det
det d(Pt /et )
dPt
=
+
+
,
Pt
Pt /et
et
et Pt /et
with the ratio Pt /et given by
Pt
et
=
ωtI (cIt
1
= At (1 + At ξt ∆∗ ) + O(γ 2 ).
II
− cII
t ) + ct
48
(A.38)
Then,
dPt
Pt
=
¡
¢
det
+ (∂t log At + ξt ∆∗ ∂t At ) dt + (γI − γII )At ξt dωtI∗ + σe (Xt )dhZ e , ω I∗ it
et
+At ∆∗ (dξt + σe (Xt )dhZ e , ξit ) + (γI − γII )At dhω I∗ , ξit + O(γ 2 ).
(A.39)
Since the bracket processes can be written as
dhZ e , ω I∗ it
dhZ e , ξit
dhω I∗ , ξit
∂ωtI∗
∂ω I∗
dt + et σe (Xt ) t dt,
∂Xt
∂et
∂ξt
= ρeX σX (Xt )
dt,
∂Xt
∂ξt ∂ωtI∗
∂ξt ∂ωtI∗
= et ρeX σe (Xt )σX (Xt )
dt + σX (Xt )2
dt,
∂Xt ∂et
∂Xt ∂Xt
= ρeX σX (Xt )
we insert the above expressions into (A.39) and (A.38). Since in an unconstrained homogeneous loginvestor economy dωtI = O(γ)dt + O(γ)dZtX , the instantaneous drift and the instantaneous volatility of
the cumulative price process, (dPt + et dt)/Pt , follow as claimed.
¤
Proof of Proposition 7
We assume without loss of generality γI < γII and that the less risk averse investor II is binding. Market
clearing implies
µ
ωtI
α(χ, t) − r(χ, t)
(1 + γI ) + γI ηt
kσ(χ, t)k2
¶
+ (1 − ωtI )wb+ (χ, t) = 1.
Solving for the interest rate and expanding in τ and γ, we obtain the interest rate r(χc , t) as given in
the proposition with
µ
¶
√
1 √
1
h γI ; √ , τ , τ = √ %τ,0 + τ %τ,1 + τ %τ,2 + γI %γ ,
τ
τ
49
where
%τ,0
=
%τ,1
=
%τ,2
=
%γ
=
´
√
1 − ωtIc ³
v + BωtIc ,
Ic
2 − ωt
³ ³
(1 − ωtIc )ωtIc kσ(χ, t)k 2B α(χc , t) −
kσ(χ, t)k
−
1
At
−
kσ(χ,t)k2
2
´
´
+ v 2 kσ(χ, t)k2 (ωtIc − 2)
2(BωtIc )3/2
,
vkσ(χ, t)k(ωtIc − 2)
%τ,1 ,
B
kσ(χ, t)k2 (1 + ηtc )
−
ωtIc − 2
¡
¢
Ic
ωt (1 − ωtIc )vkσ(χ, t)k2 2 log(1 − β)(ωtIc − 2) + (B(1 + ηtc ) − v 2 (ωtIc − 2))ωtIc )
−
(ωtIc − 2)(BωtIc )3/2
³
p
¡
¢´
Ic
Ic 2
Ic
2 Ic
Ic
1 kσ(χ, t)k(1 − ωt ) Bvωt − Bωt 2N + (ωt − 2)(v ωt − 2 log(1 − β))
−√
.
τ
B(ωtIc − 2)2 ωtIc
To obtain the price dynamics we can use (A.39). By inspection of (A.39) we see that we need to
calculate dωtIc only up to O(γ, τ ), since the expression involving ω-terms is pre-multiplied by the first
order difference γI − γII . This gives
c
e
c
ωc
dωtIc = µωc
0 (χ , t)dt + σ0 (χ , t)dZt + O(γ, τ ),
(A.40)
where
c
µωc
0 (χ , t) =
σ0ωc (χc , t) =
¢
¢
ωtIc (1 − ωtIc )2 ¡ ¡
At EC 2 (C + vσe (Xt )) − D2 vσe (Xt ) − 2C 2 σe (Xt )2 (C + vσe (Xt ))
σe (Xt )2 C 3 At
¢
ω Ic (1 − ωtIc )2 ¡ 2
+ t
%τ,0 − 2v%τ,0 σe (Xt ) + 2σ(Xt ) (v(C + vσ(Xt )) − σe (Xt ) log(1 − β))
σe (Xt )2 τ
2 (C + vσe (Xt )) ωtIc (1 − ωtIc )2
√
+
D,
(A.41)
Cσe (Xt )2 τ
µ
¶
ω Ic (1 − ωtIc ) D C + vσe (Xt )
√
− t
,
(A.42)
+
σe (Xt )
C
τ
50
with
C
=
D
=
E
=
q
%2τ,0 − 2v%τ,0 σe (Xt ) + σe (Xt )2 (v 2 − 2 log(1 − β)),
Ã
Ã
!!
v(1 − ωtIc )
2
σe (Xt ) %τ,0 + (vσe (Xt ) − %τ,0 ) 1 − p
,
ωtIc B
Ã
Ã
Ã
!2
!!
v(1 − ωtIc )
v(1 − ωtIc )
4
2
2
σe (Xt ) 1 − p
+ 2σe (Xt ) δ + µe (Xt ) + σe (Xt ) 1 − p
ωtIc B
ωtIc B
+2%τ,1 (%τ,0 − vσe (Xt )) .
We note that the dZtX term for the dynamics of ωtIc vanishes in O(γ, τ 2 ). To obtain the price dynamics,
we go back to equation (A.39). Moving from the unconstrained to the constrained economy adds
additional terms, namely
¡
¢
(γI − γII )At ξt dωtI∗ + σe (Xt )dhZ e , ω Ic it + (γI − γII )At dhξ, ω Ic it + O(γ 2 , τ 2 )
µµ
µ
¶
¶
¶
∂ξt /∂X
ωc
c
ωc
c
ωc
c
e
= (γI − γII )At ξt
µ0 (χ , t) + ρeX σX (Xt )
+ σe (Xt ) σ0 (χ , t) dt + σ0 (χ , t)dZ
ξt
+O(γ 2 , τ 2 ).
In the unconstrained economy all terms of order O(γ 2 , τ 2 ) can be omitted, but this is not true in the
constrained economy. Incorporating this term into the price dynamics, and noting from (A.39) that the
price dynamics must be of the form
dPt + et dt
= α(χc , t)dt + O(γ)dZt + σe (Xt )dZte ,
Pt
we obtain the drift and volatility as claimed in the proposition. Inserting these expressions together
with r(χc , t) into the dynamics of ωtI , we obtain the expressions for ωtIc . Finally, the explicit functions
c
ωc
c
ωc
c
ωc
c
for µωc
γ (χ , t), µτ (χ , t), σγ (χ , t), and στ (χ , t) follow. Since these are lengthy expressions they are
not displayed but can be obtained by the authors.
51
¤
Notes
1
More precisely, they simplify the structure of the dynamic optimal policies in their multiperiod model by
using some approximations of a sequence of single-period optimizations problems.
2
For an overview on VaR see Jorion (1997) and Duffie and Pan (1997).
3
The VaR limit (6) takes into account the fact that successful traders typically see their VaR limit increase
with wealth. Other forms of VaR limits may be considered, as for instance a constant limit or a limit defined
by a sum of fixed proportions of initial wealth and a running gain part (see Cuoco, He, and Issaenko (2001)).
4
Other forms of the VaR limit lead in general to wealth dependent VaR boundaries under the above approx-
imation procedure.
5
If higher-order approximations of the initial VaR constraint are considered, we obtain a sequence of control
problems (Pn )n=1,2,... which eventually converges to the original problem (P). We do not consider this issue
further in this paper.
6
One can show that solution of problem (P1) exists, i.e. the gradient of the value function is positive and
the Hessian is strictly negative definite.
7
Detemple, Garcia, and Rindisbacher (2003) combine simulation with the computation of Malliavin deriva-
tives to solve for the optimal portfolio strategy. Related to their work is the paper by Cvitanic, Goukasian,
and Zapatero (2003) who propose a method based solely on Monte Carlo simulation. Both approaches are
restricted to a complete market model setting. In our exercise we follow the approach proposed by Cvitanic,
Goukasian, and Zapatero (2003). For the estimation, we partition the investor’s time-horizon into 100 time
steps. In each time step, we draw 1000 000 random numbers using antithetic variates and calculate the paths of
the state variable X in order to calculate the evolution of price and variance.
8
Other specifications produced very similar findings as those presented in Figure 1 below.
9
By llA we denote the Heaviside function which equals one if A is true and zero otherwise.
10
This value holds for a model where volatility follows a lognormal process. For other processes, similar values
were found.
11
For γ < 0, the following conclusions would just change sign.
12
The model can easily be extended to an arbitrary number of banks.
52
References
Acerbi, C., and D. Tasche, 2001, “On the coherence of Expected Shortfall,” Working paper,
Zentrum Mathematik (SCA), TU München.
Anderson, T., L. Benzoni, and J. Lund, 2002, “An Empirical Investigation of Continuous-Time
Equity Return Models,” Journal of Finance, 57, 1239–1284.
Basak, S., and A. Shapiro, 2001, “Value-at-Risk-Based Risk Management: Optimal Policies
and Asset Prices,” Review of Financial Studies, 14, 371–405.
Berkelaar, A., P. Cumperayot, and R. Kouwenberg, 2002, “The Effect of VaR Based Risk
Management on Asset Prices and the Volatility Smile,” European Financial Management,
8, 139–164.
Black, F., 1976, “Studies in Stock Price Volatility Changes,” in Proceedings of the 1976 Business and Economic Statistic Section . pp. 177–181, American Statistical Association, Alexandria, VA.
Christie, A., 1982, “Stochastic Behavior of Common Stock Variances,” Journal of Financial
Economics, 10, 407–432.
Cuoco, D., H. He, and S. Issaenko, 2001, “Optimal Dynamic Trading Strategies with Risk
Limits,” Working paper, Wharton School, University of Pennsylvania.
Cvitanic, J., L. Goukasian, and F. Zapatero, 2003, “Monte Carlo simulation of optimal portfolios in complete markets,” Journal of Economic Dynamic and Control, 27, 971–986.
Danielsson, J., H.-S. Shin, and J.-P. Zigrand, 2001, “Asset Price Dynamics with Value-atRisk Constrained Traders,” Working paper, Department of Accounting and Finance, and
Financial Markets Group, London School of Economics.
Danielsson, J., and J.-P. Zigrand, 2001, “What happens when you regulate risk? Evidence
from a simple equilibrium model,” Working paper, Department of Accounting and Finance,
and Financial Markets Group, London School of Economics.
53
Detemple, J., R. Garcia, and M. Rindisbacher, 2003, “A Monte Carlo Method for Optimal
Portfolios,” Journal of Finance, forthcoming.
Duffie, D., and J. Pan, 1997, “An Overview of Value at Risk,” Journal of Derivatives, 4, 7–49.
Heston, S., 1993, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, 6, 341–359.
Jorion, P., 1997, Value-at-Risk: New Benchmark for Controlling Market Risk, McGraw-Hill.
Kogan, L., and R. Uppal, 2001, “Risk aversion and optimal portfolio policies in partial and
general equilibrium economies,” Working paper, University of British Columbia, Vancouver.
Schwert, G., 1989, “Why does Stock Market Volatility Change over Time?,” Journal of Finance, 44, 1115–1153.
Trojani, F., and P. Vanini, 2001, “Perturbative Solutions of Hamilton Jacobi Bellman Equations in Non-Homothetic Robust Decision Making,” Working paper, University of Southern
Switzerland, Lugano.
54